A Novel Approach to Model Error Modeling using the - - PowerPoint PPT Presentation

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A Novel Approach to Model Error Modeling using the Expectation-Maximization Algorithm

Ramón A. Delgado , Graham C. Goodwin, Rodrigo Carvajal, Juan C. Agüero

School of Electrical Engineering and Computer Science, The University of Newcastle, Australia.

December 2012

Delgado, Goodwin, Carvajal, Agüero (UoN) CDC 2012, Maui, Hawaii December 2012 1 / 19

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Motivation Robustness is one of the key ideas in control

y

G

u

∆

w z

and the basis of robust control theory.

Delgado, Goodwin, Carvajal, Agüero (UoN) CDC 2012, Maui, Hawaii December 2012 2 / 19

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Motivation Robustness is one of the key ideas in control

y

G

u

∆

w z

and the basis of robust control theory. Where does G and ∆ come from?

Delgado, Goodwin, Carvajal, Agüero (UoN) CDC 2012, Maui, Hawaii December 2012 2 / 19

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Motivation Robustness is one of the key ideas in control

y

G

u

∆

w z

and the basis of robust control theory. Where does G and ∆ come from? This talk addresses this problem.

Delgado, Goodwin, Carvajal, Agüero (UoN) CDC 2012, Maui, Hawaii December 2012 2 / 19

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Outline Explanation of model error description. Some previous approaches. Presentation of our approach to model error description. A numerical example.

Delgado, Goodwin, Carvajal, Agüero (UoN) CDC 2012, Maui, Hawaii December 2012 3 / 19

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Main contribution Maximum Likelihood Estimate Provide estimates for the nominal model and model error that correspond to the maximum likelihood estimate.

Delgado, Goodwin, Carvajal, Agüero (UoN) CDC 2012, Maui, Hawaii December 2012 4 / 19

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Model Error Description In System Identification the aim is to provide a nominal model that tries to describe the dynamics of the true system. yt = G(q)ut + vt (1)

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Figure: Magnitude Bode plot.

Delgado, Goodwin, Carvajal, Agüero (UoN) CDC 2012, Maui, Hawaii December 2012 5 / 19

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Model Error Description However, a systematic error occurs when the true system is more complex than the nominal model.

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Figure: Magnitude Bode plot.

Delgado, Goodwin, Carvajal, Agüero (UoN) CDC 2012, Maui, Hawaii December 2012 6 / 19

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Model Error Description Our aim is to provide a model error description corresponding to the nominal model.

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Figure: Magnitude Bode plot.

Delgado, Goodwin, Carvajal, Agüero (UoN) CDC 2012, Maui, Hawaii December 2012 7 / 19

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Some Previous Approaches Set Membership. Stochastic embedding. Model Error Modelling. and several others, as the based on finite sample properties, or in resampling, sub-sampling and/or bootstrapping techniques.

Delgado, Goodwin, Carvajal, Agüero (UoN) CDC 2012, Maui, Hawaii December 2012 8 / 19

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Previous Approach: Set Membership Deal with the problem of bounded but unknown errors. Given a bound for the error, Set Membership delivers a Feasible System Set. Feasible System Set = {G ∈ H : y − Gu < δ} Deterministic framework. Finding the solution set can be difficult depending on the model structure and the norm used to bound the error. The value of δ must be specified a priori. Focus into the “worst-case” model.

Delgado, Goodwin, Carvajal, Agüero (UoN) CDC 2012, Maui, Hawaii December 2012 9 / 19

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Previous Approach: Stochastic Embedding The nominal model structure is embedded into a larger class of model, where the model error is characterised as a realization of a stochastic process. ¯ G = G · (1 + G∆) (2) Probabilistic framework. Solved by using Least Squares. The nominal model is not estimated simultaneously with the model error. A Random Walk in the frequency domain was proposed as a description for the model error.

Delgado, Goodwin, Carvajal, Agüero (UoN) CDC 2012, Maui, Hawaii December 2012 10 / 19

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Previous Approach: Model Error Modelling The nominal model and the model error description are obtained in two steps. In the first step the nominal model is obtained by using any standard tool. In the second step, the residues ε = y − ˆ y are used to obtain the model error description. yt = G(q)ut + vt (3) ˆ yt = ˆ G(q)ut (4) εt = yt − ˆ yt (5) εt = F(q)ut + H(q)wt (6) Probabilistic framework Allow the use of standard tools of system identification to estimate the nominal model, as well as the model error. The model error should have a complex structure.

Delgado, Goodwin, Carvajal, Agüero (UoN) CDC 2012, Maui, Hawaii December 2012 11 / 19

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Some issues with previous approaches In Set membership identification, the upper bound δ needs to be specified a priori. This could be challenging for some specific problems. In both probabilistic methods, i.e. Stochastic Embedding and Model Error Modelling, the estimated nominal model is not affected by the model error description. In Stochastic Embedding the nominal model complexity is limited by the use of Least Squares.

Delgado, Goodwin, Carvajal, Agüero (UoN) CDC 2012, Maui, Hawaii December 2012 12 / 19

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Some issues with previous approaches In Set membership identification, the upper bound δ needs to be specified a priori. This could be challenging for some specific problems. In both probabilistic methods, i.e. Stochastic Embedding and Model Error Modelling, the estimated nominal model is not affected by the model error description. In Stochastic Embedding the nominal model complexity is limited by the use of Least Squares.

Delgado, Goodwin, Carvajal, Agüero (UoN) CDC 2012, Maui, Hawaii December 2012 12 / 19

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Some issues with previous approaches In Set membership identification, the upper bound δ needs to be specified a priori. This could be challenging for some specific problems. In both probabilistic methods, i.e. Stochastic Embedding and Model Error Modelling, the estimated nominal model is not affected by the model error description. In Stochastic Embedding the nominal model complexity is limited by the use of Least Squares.

Delgado, Goodwin, Carvajal, Agüero (UoN) CDC 2012, Maui, Hawaii December 2012 12 / 19

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Some issues with previous approaches In Set membership identification, the upper bound δ needs to be specified a priori. This could be challenging for some specific problems. In both probabilistic methods, i.e. Stochastic Embedding and Model Error Modelling, the estimated nominal model is not affected by the model error description. In Stochastic Embedding the nominal model complexity is limited by the use of Least Squares. Our contribution Estimate both, the nominal model and the model error description

• simultaneously. Moreover, this estimate correspond to a Maximum

Likelihood Estimate.

Delgado, Goodwin, Carvajal, Agüero (UoN) CDC 2012, Maui, Hawaii December 2012 12 / 19

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Proposed approach (PA) The nominal model structure is embedded into a larger class of

• model. The estimation of the nominal model and the model for the

error is made simultaneously. Probabilistic framework. Simultaneous estimation of the nominal model and the model error description. The complete estimated model (containing the nominal model and the model error description) correspond to the Maximum Likelihood Estimate. Allow more complex structures than Stochastic Embedding for the nominal model. (By using Expectation-Maximization algorithm.)

Delgado, Goodwin, Carvajal, Agüero (UoN) CDC 2012, Maui, Hawaii December 2012 13 / 19

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Model description example Consider the following description of the model G(jωk) = Go(jωk)(1 + G∆(jωk)), (7) and the corresponding data generating system Y(jωk) = G(jωk)U(jωk) + V(jωk), (8) where ωk = 2πk

N , Y(jωk) and U(jωk) are the Discrete Fourier

Transform of the measurement and the input signal, respectively. Moreover, V(jωk) is a realization of a zero mean proper Gaussian process and G∆(jωk) is a realization of a Random Walk process in the frequency domain .

Delgado, Goodwin, Carvajal, Agüero (UoN) CDC 2012, Maui, Hawaii December 2012 14 / 19

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Comparison Example 2 (not in the paper) Consider the following data generating system y(t) = q−1 + 0.5q−2 1 − 2.2q−1 + 2.42q−2 − 1.87q−3 + 0.7225q−4 u(t) + w(t) (9) where u(t) is a Pseudo Random Binary Signal (PRBS), and w(t) is chosen to be one of the signal registered during the Loma Prieta earthquake, in October, 19891. The nominal model to estimate is a second order Output Error model.

1This signal has been made available by The MathWorks Inc. 2Wolfgang Reinelt. i4c: Identification for control package (version 1.1b5).

Linköping University, Linköping, Sweden, Sept. 2000. http://www.wolfgang-reinelt.de/i4c/

Delgado, Goodwin, Carvajal, Agüero (UoN) CDC 2012, Maui, Hawaii December 2012 15 / 19

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Comparison Example

Output composition

200 400 600 800 1000 −40 −20 20 40 Noise−free Output Samples

200 400 600 800 1000 −100 −50 50 100 disturbance Samples

SNR = 1

Delgado, Goodwin, Carvajal, Agüero (UoN) CDC 2012, Maui, Hawaii December 2012 16 / 19

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Comparison Example

Results for 99% confidence

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(a) Model Error Modeling

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(b) Set Membership

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(c) PA with constant variance un- dermodelling, with Nexp = 1.

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(d) PA RW undermodelling with Nexp = 25.

Delgado, Goodwin, Carvajal, Agüero (UoN) CDC 2012, Maui, Hawaii December 2012 17 / 19

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Summary We develop a new method to provide a model error description. The developed method allow to estimate the nominal model and the model error description, simultaneously. The complete model (containing the nominal model and the model error description) correspond to the Maximum Likelihood Estimate.

Delgado, Goodwin, Carvajal, Agüero (UoN) CDC 2012, Maui, Hawaii December 2012 18 / 19

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Thanks for your attention! Any questions?

Delgado, Goodwin, Carvajal, Agüero (UoN) CDC 2012, Maui, Hawaii December 2012 19 / 19

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Wolfgang Reinelt. i4c: Identification for control package (version 1.1b5). Linköping University, Linköping, Sweden, Sept. 2000. http://www.wolfgang-reinelt.de/i4c/.

Delgado, Goodwin, Carvajal, Agüero (UoN) CDC 2012, Maui, Hawaii December 2012 20 / 19

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Numerical example Consider the nominal model Go(z) = 0.12 z2 − 1.3z + 0.42, (10) and G∆(jωk) having variance σ2

w = 4 · 10−4, and V(jωk) having

variance σ2

v = 0.1.

We run Nexp = 20 different experiments each with L = 50 samples , with the same input signal {Uk}L−1

k=0, generated as a proper Gaussian

sequence with zero mean and variance σ2

u = 4.

On each experiment we have different realizations of the noise and the uncertainty.

Delgado, Goodwin, Carvajal, Agüero (UoN) CDC 2012, Maui, Hawaii December 2012 20 / 19

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Model realizations Estimated model

The estimated model is given by G(ˆ β, z) = 0.116/(z2 − 1.307z + 0.423), ˆ σ2

w = 0.0033, and ˆ

σ2

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Delgado, Goodwin, Carvajal, Agüero (UoN) CDC 2012, Maui, Hawaii December 2012 21 / 19

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Expectation-Maximization Algorithm The complete data is composed by the measured data Y and also an unmeasured data X, known as the hidden variables. E-step Q(θ, ˆ θi) = E{log p(Y, X|θ)|Y, ˆ θi}. (11) M-step ˆ θi+1 = arg max

θ∈Ω Q(θ, ˆ

θi). (12)

Delgado, Goodwin, Carvajal, Agüero (UoN) CDC 2012, Maui, Hawaii December 2012 22 / 19

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Random walk realizations

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Random walk realizations Evolution in time

Delgado, Goodwin, Carvajal, Agüero (UoN) CDC 2012, Maui, Hawaii December 2012 23 / 19